In this article Schoenfeld compares and contrasts two different post-secondary classes that he conducts: an undergraduate mathematical problem solving course, and a graduate level mathematics education research group. Schoenfeld sets the stage by acknowledging how differently his two courses are structured. In the problem solving course, he has control over what is taught, learned, and ultimately the steps the students take to get the answers; whereas, in his research group it is more student directed, he and the students engage in joint work together to find answers to both their questions and his.
Ultimately his main point is that he builds community in both situations which allows for engaged inquiry. This could be easily seen for me in the graduate level mathematics course, where Schoenfeld often finds himself in an area of unfamiliar research and is working with his graduate students to discover new territories. I was more intrigued with how he did it in his problem-solving course. The three main things that Schoenfeld acknowledge as crucial are:
"a. There is a common understanding that we are all seeking a particular kind of knowledge, and that while some of us know more than others, 'answers' are not generally known in advance.
b. The real 'authority' is not the Professor - it's a communally accepted standard for the quality of explanations, and our sense of what's right ...
c. There is a feeling of trust, in that we must feel free to have our ideas (and not ourselves) compete ..."
I have always realized the need to build a problem solving environment in the classroom where the students feel it is safe to explore, experiment, and ultimately work together to find the solution to problems, but I had never thought really thought of my role as the teacher in the end when we had developed that community. I love the idea that the students ultimately don't need the teachers' authority to tell them if their answers are right or wrong, that they will be able to "internalize the standards of mathematical judgment for themselves." To do so, Schoenfeld makes reference to John Mason's framework: "First convince yourself. Then convince a friend. Finally, convince an enemy." Schoenfeld goes into great details of his ability to teach and release the students safely into this community atmosphere. The problems he chooses are clearly purposeful and allow for further questions when the "initial answer" is found. I would love to ask the author how he would approach the student who is reluctant to find multiple solutions or defend their answer. Ultimately, I guess you are hoping that the community atmosphere will prevail, however, I have found that it can often take only one to ruin the atmosphere in the room.
Lastly, I am left excited by the fact that lots of schools are currently making a conscious switch to trying to do more mathematical problem based learning and I feel that if this inquiry starts at a young age, the sky is the limits with where their thinking may go!
b. The real 'authority' is not the Professor - it's a communally accepted standard for the quality of explanations, and our sense of what's right ...
ReplyDeleteI love this. This one standard gives rise to a wealth of implications in the classroom. In the, typically black and white, subject that is high school mathematics there is a culture devoted to finding the answer to a problem that is supported by assessments that highlight students' answers. When the professor knows 'the answer' then students aren't thinking for themselves, rather, they are trying to find out what their professor is thinking. This culture lends itself to a focus on the product of mathematics, rather than thinking of mathematics as a procedure, or better yet, a way of thinking.
This one standard by Schoenfeld however wipes away much of this black-and-white mathematics. By removing the supposed purpose of mathematics from trying to simply find 'the answer' students are instead now responsible for defending their mathematics, and being able to prove to their peers that their answer is the correct one.
It is interesting to note how the author strives and succeeds to create a meaningful community of learners in both settings. In trying to accomplish this, the author creates a less artificial setting. I think that this is the hardest to accomplish in a high school mathematics classroom, where students are not only competing with each others' ideas, but they are also competing (intellectually, individually, and collectively) with each other. Another interesting aspect of the article is the role of the professor.
ReplyDeleteThe author's students from his problem solving class and his research group were a part of the setting. His students chose to be there and they implicitly accepted the author as an authority figure. In a secondary setting, this may not be the case. And so trying to replicate the author's model of creating a community of learners may not work in a high school context. In the existing competitive (e.g. for admission to universities) atmosphere, I doubt that this model would work. I can think of two reasons for my doubts. First, How can mathematics teachers in high school settings be both coaches and judges at the same time to create a feeling of trust? Second, would the teachers possess a wide range of problem solving skills and strategies to cater to diverse learners?